To solve the expression $-5 \frac{2}{3} + 3 \frac{1}{4} + (-7 \frac{1}{3})$, we need to follow these steps:

1. Convert Mixed Numbers to Improper Fractions:

   • For $-5 \frac{2}{3}$: $-5 \frac{2}{3} = -\left(5 + \frac{2}{3}\right) = -\frac{15}{3} - \frac{2}{3} = -\frac{17}{3}$

   • For $3 \frac{1}{4}$: $3 \frac{1}{4} = 3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4}$

   • For $-7 \frac{1}{3}$: $-7 \frac{1}{3} = -\left(7 + \frac{1}{3}\right) = -\frac{21}{3} - \frac{1}{3} = -\frac{22}{3}$

2. Find a Common Denominator:

   The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. We convert each fraction to have a denominator of 12:

   • For $-\frac{17}{3}$: $-\frac{17}{3} = -\frac{17 \times 4}{3 \times 4} = -\frac{68}{12}$

   • For $\frac{13}{4}$: $\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}$

   • For $-\frac{22}{3}$: $-\frac{22}{3} = -\frac{22 \times 4}{3 \times 4} = -\frac{88}{12}$

3. Add the Fractions Together:

   Now, we add the fractions:

   $-\frac{68}{12} + \frac{39}{12} + \left(-\frac{88}{12}\right) = \frac{-68 + 39 - 88}{12} = \frac{-117}{12}$

4. Simplify the Fraction:

   Simplify $\frac{-117}{12}$ by finding the greatest common divisor (GCD) of 117 and 12. The GCD is 3:

   $\frac{-117 \div 3}{12 \div 3} = \frac{-39}{4}$

5. Convert Back to a Mixed Number:

   $\frac{-39}{4}$ can be written as a mixed number:

   $-9 \frac{3}{4}$

Final Answer:

$-5 \frac{2}{3} + 3 \frac{1}{4} + (-7 \frac{1}{3}) = -9 \frac{3}{4}$

Would you like more details or have any questions?

Related Questions:

1. How do you find the least common multiple (LCM) of two numbers?
2. How can improper fractions be converted back to mixed numbers?
3. What is the process for finding the greatest common divisor (GCD)?
4. How do you simplify a complex fraction?
5. What are other methods to handle mixed numbers in addition and subtraction?

Tip:

Always make sure to convert all mixed numbers to improper fractions first when dealing with addition or subtraction.